A SUBSOLUTION THEOREM FOR THE MONGE-AMP`ERE EQUATION OVER AN ALMOST HERMITIAN MANIFOLD*

School of Mathematical Sciences,University of Science and Technology of China,Hefei 230026,China

E-mail :zjgmath@ustc.edu.cn

Abstract Let Ω ?M be a bounded domain with a smooth boundary ?Ω,where (M,J,g)is a compact,almost Hermitian manifold.The main result of this paper is to consider the Dirichlet problem for a complex Monge-Amp`ere equation on Ω.Under the existence of a C2-smooth strictly J-plurisubharmonic (J-psh for short) subsolution,we can solve this Dirichlet problem.Our method is based on the properties of subsolutions which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds.

Key words complex Monge-Amp`ere equation,almost Hermitian manifold,a priori estimate,subsolution,J-plurisubharmonic

1 Introduction

Let (M,J,g) be a compact almost Hermitian manifold of real dimension 2n,and let Ω ?Mbe a smooth domain with a smooth boundary?Ω.In what follows,we denote byωthe Khler form ofg,i.e.,

for all smooth vector fieldsX,YonM.We shall consider the subsolution theorem for the Monge-Amp`ere equation

Our main result is

Theorem 1.1Letφ,h∈C∞() with infh ＞-∞.Suppose that there exists a strictlyJ-psh subsolution∈C2() for eq.(1.1),that is,

Then there exists a unique smooth strictlyJ-psh solutionufor eq.(1.1).

The study of the complex Monge-Amp`ere equation (1.1) (on Cn) is closely related to certain problems in geometry and complex analysis;see,for instance,[7,13,16] and references therein.The equation has been studied extensively over the past several decades;see [2–5,10,13,14,16,18,22–24,28,32,34] etc.Inspired by Guan’s work [13],it is natural to assume the existence of subsolutions in order to solve eq.(1.1).

The purpose of this paper is to study the Dirichlet problem for the complex Monge-Amp`ere equation on a general manifold,where the almost complex structure might not be integrable;that is,a manifold,locally,does not look like Cn.Let us remind ourselves that when the domain Ω ?Madmits a strictlyJ-psh defining function,the eq.(1.1) was already solved by Pli′s [27].His resolution could be understood as a generalized version of [5],but the underlying structure is only almost complex.Many interesting results were also obtained by Harvey-Lawson [17].

The Dirichlet problems regarding other related geometric PDEs also attracts the attension of many mathematicians.For instance,Wang-Zhang [35] studied the Dirichlet problem for the Hermitian-Einstein equation over an almost Hermitian manifold.In addition,the twisted quiver bundle on an almost complex manifold was researched by Zhang [36].Very recently,Li-Zheng [25] investigated the Dirichlet problem for a class of fully nonlinear elliptic equations,and obtained the boundary second order estimates.In the aspect of real case,one can also refer[1,20].

The structure of this paper is as follows: in Section 2 we collect some basic concepts regarding almost Hermitian manifolds.In Sections 3–5 we give the global estimates up to the second order.Once we have these estimates in hand,higher order estimates can be also obtained by the classical Evans-Krylov theory (see,for instance,[32]) and the Schauder theory.Then we can use the standard continuity method to obtain the existence;the proof of this can be found in [13],so we shall omit the standard step here.In Section 6,we obtain a strictlyJ-psh subsolution for (1.1) under the existence of a strictlyJ-psh defining function.

2 Preliminaries

Let (M,J,g) be a compact manifold of real dimension 2nwith the Riemannian metricgsatisfying that

whereJis the almost complex structure.Then the complexified tangent bundle can be divided as

whereT0,1MandT1,0Mare theofJ.Similarly,the induced almost complex structureJ*on the cotangent bundleT*Mis defined byJ*α:=-α°J.Then we have a natural decomposition

For brevity,we will also denoteJ*byJ,if no confusion occurs.For the decomposition of thek-th product of a complexified contangent bundle,

Let Ap,qbe the set of smooth sections on Λp,qMand denote that

We consider the exterior derivatived: Ak→Ak+1satisfyingd2=0.Let Πp+1,q,Πp,q+1,Πp+2,q-1and Πp-1,q+2be the projection of Ak+1to Ap+1,q,Ap,q+1,Ap+2,q-1and Ap-1,q+2,respectively.Thus,

In particular,ifv∈C2(M,R),thenv∈A0,1and

Taking the complex conjugates and adding together,

Then,in this local chart,

whereθ1,···,θnis a localg-orthonormal frame ofT1,0Mdual toe1,···,en.Thus we can rewrite the equation in (1.1) as

Let us define its linearized operator by

Definition 2.1For anyv∈C2(M,R) and with Ω ?Mbeing an open set,

(1) we say thatvisJ-psh on Ω if the matrixis nonnegative at each point of Ω;

(2) we say thatvis strictlyJ-psh on Ω if,for eachφ∈C2(Ω),there existsε0＞0 such thatu+εφisJ-psh on Ω for all 0＜ε ＜ε0.

We denote the set ofJ-psh functions on Ω by PSH(Ω).

Let us recall the notion of canonical connections on almost Hermitian manifolds.

Supposing that (M,J,g) is an almost Hermitian manifold,there exists a canonical connection ?onMwhich plays a very similar role to that of the Chern connection on the Hermitian manifold.Usually,we say that a connection on (M,J,g) is an almost-Hermitian connection if?g=?J=0.Noticing that such connection always exists [21],we have the following theorem(see [12,33]):

Theorem 2.2There exists a unique almost-Hermitian connection ?on an almost Hermitian manifold (M,J,g) whose (1,1) part of the torsion vanishes.

This connection was found by Ehresmann-Libermann [9].Sometimes it is also referred to the Chern connection,because no confusion occurs whenJis integrable.Under a local frame like the previous one,we have that

2.1 Properties of Subsolution

The following lemma is due to Guan [15],who proved it for more general fully nonlinear PDEs:

Lemma 2.3Let∈C2() be a strictlyJ-psh subsolution to the eq.(1.1).There exist constantsN,θ ＞0 such that ifat a pointp∈Ω whereand the matrixis diagonal,then

Let us remark that sinceuis strictlyJ-psh,there exists a uniform constantτ∈(0,1) such that

2.2 Maximum Principle

We have the following useful lemma.

Lemma 2.4([5,p.215]) Let Ω ?Mbe a smooth bounded domain.Ifu,v∈C2() ∩PSH(Ω) withustrictlyJ-psh and det,thenu-vattains its maximum on?Ω.

3 C0 and C1 Estimates

3.1 Uniform Estimate

Let∈C2() be a solution of the Dirichlet problem

Lemma 3.1Letu(resp.) be the solution (resp.subsolution) of eq.(1.1).We have that

ProofOn the one hand,asis a subsolution of (1.1),the first inequality follows from Lemma 2.4.On the other hand,sinceL(u)=n,we know thatuis a subsolution of (3.1).By the maximum principle (for operatorL),we also get the second inequality. □

3.2 Boundary Gradient Estimate

Lemma 3.2Letu(resp.u) be a solution (resp.subsolution) of eq.(1.1).Then there exists a constantsuch that

ProofBy the previous lemma,together with the fact thatu,uandhhave the same boundary valueφ|?Ω,we haveon?Ω,and the lemma follows. □

3.3 Global Gradient Estimate

Proposition 3.3Letu(resp.) be a solution (resp.subsolution) of eq.(1.1).Then

1The constants C,C′ in the rest of the section are distinct,where C is a constant depending on all the allowed data,but C′ further depends on a constant B that we are yet to choose.By a straightforward calculation,

Differentiating (2.2) alongej,

Notice that

We may assume that |?u| ?1 (otherwise we are done),and set

By the Cauchy-Schwarz inequality,for each 0＜ε≤,

It then follows from (3.5) that

As 0＜ε≤,1 ≤(1 -ε)(1+2ε).Thus,

It follows from (3.13) and (3.14) that

Combining (3.5),(3.12) and (3.15) gives us

Case 1for some N as in Lemma 2.3.We divide the proof into two parts.

Subcase 1(i)Iffor somej,whereD ＞0 is a large constant to be determined shortly,then

We may assume that |?u| ≥|?u|,whence |?η| ≤2|?u|.Thus,

Substituting this into (3.16),

Substituting this into (3.18),we get that |?u| ≤C′.

The fact thatis strictlyJ-psh implies that

It follows from (3.16),(3.17),(3.19) and (3.20) that

which implies |?η| ≤C′,whence (3.4) follows. □

4 Interior C2 Estimate

In this section we follow the arguments of [8] to estimate the largest eigenvalueλ1(2u) of the real Hessian2u,whereis the Levi-Civita connection onM.

Theorem 4.1Letu(resp.u) be a solution (resp.subsolution) of eq.(1.1).We have

We may assume that Ω′is a nonempty (relative) open set,otherwise we are done.Aszapproaches?Ω′?Ω,Q →-∞,if Q achieves its maximum on?Ω,then we are done,by (4.1).Thus,we may assume that Q achieves its maximum in Int(Ω′).Nearx0,we choose a localg-unitary frame (e1,···,en) such that,atx0,

wheregαβ:=g(?α,?β).

for some smooth sectionSonT*M?T*Msuch that

with the equality only atx0,but alsoλ1(Φ) ∈C2(Ω) (cf.[8,11]).For anyβ,letVβbe eigenvector of Φ with an eigenvalueλβ.The proof needs the following derivatives ofλ1,which can be found in [8,11,29]:

Lemma 4.2Atx0,we have that

We will prove (4.1) by applying the maximum principle to the quantity

Clearly,Qattains its maximum atx0.Thus,atx0,

For the rest of this section we may assume thatfor the constantNin Lemma 2.3(otherwise we are done).

4.1 Lower Bound of L(Q)

Proposition 4.3For eachε∈(0,],atx0,we have that

ProofFirst,we calculateL(λ1).Let

By Lemma 4.2 and (4.4),we can infer that

ApplyingV1to eq.(2.2) twice,

Lemma 4.4Ifλ1?1,then

ProofBy a direct calculation,

Then the lemma follows from (4.10) ifλ1?1. □

It follows from (4.9) and (4.11) that

By (3.12),we have that

Lemma 4.5For each 0＜ε≤1/2,we have that

ProofAssume that

whereμiβ∈C are uniformly bounded constants.Then,

where in the last inequality we have used is;see [8].HereTis the torsion vector field of (g,J) [31,p.1070].It follows from the above three inequalities that

Then,by (4.16),we obtain (4.15). □

Consequently,Proposition 4.3 follows from (4.14)–(4.15). □

4.2 Proof of Theorem 4.1

We divide the proof into three cases.

Case 1Atx0,

Case 2Atx0,

Substituting this into (4.8),

Proof of Case 1SinceL() is uniformly bounded from below,it follows from the concavity ofLthat

2In what follows,CB are positive constants depending on B.Notice thatare pairwisely comparable,by (4.20),so

Thus the complex covariant derivatives

and this proves (4.1).

Proof of Case 2It follows from (4.8) and (4.21) that

Using the fact thatL() ≥θ(1+U) (by (2.4)),we have that

which yields a contradiction if we further assume thatBis large enough.

Case 3If the Cases 1 and 2 do not hold,we define

Clearly,1 ∈I,nI.Hence,we may letI={1,2,···,p} for a certainp ＜n.

Lemma 4.6Assume thatB≥6nsup(||2).Atx0,we have

ProofIt follows from (4.6) and the inequalitythat

Let us define a new (1,0) vector field by

Atx0,there exist?1,···,?n∈C such that

Lemma 4.7Atx0,for allkI.

ProofThe proof is from [8];we include it here for the convenience of the reader.Now we have

This proves the lemma. □

Now we estimate the first three terms in Proposition 4.3.SinceJV1isg-unitary andg-orthogonal toV1,there existμ2,···,μ2n∈R such that

Lemma 4.8Atx0,for any constantγ ＞0,

ProofWe divide the proof into three steps.

We have the first term is

whereO(λ1) are those terms which can be controlled byλ1.The second term is

Step 2It follows from (4.25) and Lemma 4.7 that

By the Cauchy-Schwarz inequality,

With these,for eachγ ＞0,

Then the lemma follows from (4.26),(4.29) and (4.30). □

ProofIt suffices to prove that

We divide the proof into two assumptions. □

Assumption 1Atx0,we assume that

ProofTaking this,as well as Lemma 4.8,we get that

Assumption 2Atx0,we assume that

ProofComputing atx0,we get that

It then follows from (4.34) that.Hence,

Substituting this into Lemma 4.8 yields that

where in the last inequality we relied on the fact that.This proves (4.31),and hence the proof of the lemma is complete. □

Now we complete the proof of the interior second order estimate.It follows from Lemma 4.9 and (4.8) that,atx0,

We chooseBsufficiently large such thatBθ≥C.This then yields a contradiction,and we have completed the proof. □

Remark 4.10The interiorC2,αestimates follow from the Evans-Krylov theorem and an extension trick introduced by Wang [34] in the study of the complex Monge-Amp`ere equation.Then the higher order estimates can be obtained by Schauder estimates.

5 Boundary C2 Estimates

In this section we shall derive the estimate

for a certain dependent constantC.

5.1 Pure Tangential Estimates

Let us fix a pointz∈?Ω,and define

Sinceu-=0 on?Ω,we can writeu=+ρσin a neighborhood ofz,whereσis a function defined on?Ω which depends,linearly on the first order derivatives ofu-.For arbitrary vector fieldsX,Ywhich are tangential to?Ω,

It follows from theC1estimate that

Then the pure tangential estimates follow by the randomicity ofz.

5.2 Mixed Direction Estimates

Proposition 5.1LetN∈TzMbe orthogonal to?Ω such thatNρ=-1,and letXbe a vector field which is tangential to?Ω.We have that

whereCdepends onand other known data.

ProofLet O ?Mbe a local coordinate chart withz∈O.We may pick up real vector fieldsX1,···,Xnwhich are tangential atzto?Ω such thatX1,JX1,···,Xn,JXnis agorthonormal local frame nearz.Furthermore,we assume thatYn:=JXnis the normal vector on?Ω nearz.

Fixing a constantδ ＞0,we set

We shall prove (5.2) by applying the maximum principle to

for a negative functionv∈C∞(Ωδ),to be determined later.Letbe a neighborhood ofz,and setSδ:=O′∩Ωδ.

First we chooseBlarge enough such thatQ±≤0 on?Sδ.We shall proveQ±≤0 inδfor a large constantA.Otherwise,suppose thatQ±attains its maximum at a pointx0∈Sδ.Lete1,···,enwith

be a localg-orthonormal frame in a neighborhood ofx0such that the matrixis diagonal atx0.

The following lemma plays a significant role in our proof:

Lemma 5.2There exist some uniform positive constantst,δandεsufficiently small,and anNsufficiently large,such that the function

ProofAsu≤uandv≤0 inδ,if we letδ?tbe small enough such thatNδ ＜t,then by a direct calculation and the property of the mixed discriminant,atx0,

where we used (2.5).It follows that

ift?1.By an elementary inequality,we deduce that

We chooseNlarge enough such that

Substituting this into (5.5),we get that.This completes the proof.□Now we continue to prove Proposition 5.1.Clearly,

For each vector fieldY,

There existαjk,βjk∈C such that

It follows that

which implies that

where in the last inequality we used the fact that12a2+2ab≥-2b2.It then follows from (5.4),(5.6) and (5.7) that

ifAis large enough such thatAε≥(B+1)C,which contradicts to the fact thatQ±attains its maximum atx0.Consequently,Q±≤0 inandQ±(z)=0.By Hopf’s lemma,|NXu|(z) ≤C.□

5.3 Pure Normal Estimates

Proposition 5.3LetN∈TzMbe orthogonal to?Ω atzsuch thatNρ=-1.We have

Before proving this,let us recall some useful facts from the matrix theory.For any Hermitian matrixwith eigenvaluesλi(A),let,and we denote the eigenvalues ofby3In what follows,we let α,β=1,2,···,n -1;i,j=1,2,···,n.It follows from Cauchy’s interlace inequality [19] and [6,p.272] that when,

To this end,we follow an idea of Trudinger [30] and set

Then we are reduced to showing

be a positive orthant in Rn-1,we divide the proof into two cases.

Case 1Assume that it holds that

By virtue of (5.1) and (5.2),we know that

where C ?Γ∞is compact.Then there existc1,R1∈R＞0depending onλ′((x0)) such that

By continuity,there exists a cone??！轪nd a neighborhood of C such that

It follows from (5.12),(5.13) and (5.14) that

which yields a contradiction to (2.2).Hence (5.10) follows by letting

Case 2Assume that it holds that

on the set of (n-1)2Hermitian matrices withλ′(E) ∈?！?Notice thatis concave and finite,since the operatorλlog det(λ) is concave and continuous.Hence,there exists a symmetric matrixsuch that

for any (n-1)2Hermitian matrixE.On?Ω,sinceu=u,

Notice that 0＜～c ＜∞,since the operatorλlog det(λ) is strictly increasing with respect to each variable.Now we divide the proof into two cases.

Subcase 2 (i)Assume that atx0,

Subcase 2 (ii)Assume that atx0,

for some uniform constantτ ＞0.We may assume thatη≥τin Ωδby shrinkingδagain if necessary.

Let us define a function in Ωδby

By a direct calculation,

It follows from (5.16) that

Thus,Φ(x0)=0 and Φ ≥0 nearx0on?Ω.Define

One can verify that Φ+Ψ ≥0 on?Ωδand

provided thatA?B?1.By Hopf’s lemma,we know thatYnΦ(x0) ≥-C,thenYnYnu(x0) ≤C.

Now we are in a position where all the eigenvalues ofU(x0) are bounded,soλ(U)(x0) is contained in a compact subset of Γn.Since the operatorλlog det(λ) is strictly increasing with respect to each variable,

whenRis large enough.This proves (5.10),and the proof is complete. □

6 Existence of Subsolutions

Suppose that Ω ?Mis a smooth pseudoconvex domain,and letρbe a strictlyJ-psh defining function for Ω.Then there exists a uniform positive constantγ ＞0 such thatFor eachs ＞0,we set

Therefore,

We may chooses?1 such that detNotice that=φon?Ω,sois a desired subsolution of eq.(1.1).

AcknowledgementsThe author would like to thank his thesis advisor Professor Xi Zhang for his constant support and advice.